# How to decrease number of noises in stochastic differential equation

I have a Quantum SDE containing both white and color noises (open quantum system). $$\dot\rho(t) = A\rho_s + (\nu_{1t}\hat{c}^\dagger \hat{X}^-_1 + \omega_{1t} \hat{X}^+_1 \hat{c})\rho_s +(\nu_{2t} \hat{X}^+_2\hat{c} -\omega_{2t}\hat{c}^\dagger \hat{X}^-_2)\rho_s$$ Here A is a constant, $$\hat{c}^\dagger$$ and $$\hat{c}$$ are creation and annihilation operators belong to system. $$X^\pm_j$$ are operators belong to fermionic environment. The order of $$X^\pm_j$$, $$\hat{c}^\dagger$$ and $$\hat{c}$$ is fixed (no commutation or anti-commutation relation exist between $$X^\pm_j$$ and $$\hat{c}^\dagger$$, $$\hat{c}$$ ). $$\nu_j$$ is white noise, $$\omega_{1t}$$ and $$\omega_{2t}$$ are defined as $$\omega_1(t) = \int_{t_0}^{t} C^+(t-\tau)\nu_{1\tau} d\tau$$ and $$\omega_2(t) = \int_{t_0}^{t} C^-(t-\tau)\nu_{2\tau} d\tau$$ Where $$C^\pm$$ is defined as
$$C^\sigma(t)=\int_{-\infty}^{\infty} d\omega e^{i\sigma\omega t} f(\omega) J(\omega)$$ My question here is that how can we decrease the number of noises by combining white noises $$\nu_j$$ and color noises $$\omega_j$$ while they are connected to different operators. I know how to decrease the number of noises if they are connected to same operators. Any idea about different operator case, any article, book ? Your help will be appreciated.

• Can you give an example how you "decrease the number of noises" for the same operator case? It might help to clarify your question. Dec 26, 2018 at 13:26
• For example, For a case where $\nu_{1t}$ and $\omega_{1t}$ are connected to same operators, $\nu_{1t}$ and $\omega_{1t}$ can be replaced by one noise, i.e , $$\dot\rho(t) = A\rho_s + (\nu_{1t}\hat{c}^\dagger \hat{X}^-_1 + \omega_{1t} \hat{c}^\dagger \hat{X}^-_1 )\rho_s +(\nu_{2t} \hat{X}^+_2\hat{c} -\omega_{2t}\hat{c}^\dagger \hat{X}^-_2)\rho_s$$ In above equation the 2nd term on R.H.S can be written as $(\nu_{1t}+ \omega_{1t} ) \hat{c}^\dagger \hat{X}^-_1 \rho_s$ . Now $(\nu_{1t}+ \omega_{1t} )$ can be replaced by another noise $\alpha_{t}$ whose auto and cross-correlation can be found. Dec 27, 2018 at 8:37

Thank you for your clarification. The best I can propose is to write the operators in terms of a quadratic form $$\left[\begin{array}{ccc} c^{\dagger} & X_{1}^{+} & X_{2}^{+}\end{array}\right]Q\left[\begin{array}{c} c\\ X_{1}^{-}\\ X_{2}^{-} \end{array}\right]\rho_{s}$$ , where $$Q$$ is a $$3x3$$ matrix that contains the $$c$$-number noise terms. as you see, there are 9 entries, one for each possible operator combination. in your example case in the comments, the form is $$\left[\begin{array}{ccc} c^{\dagger} & X_{1}^{+} & X_{2}^{+}\end{array}\right]\left[\begin{array}{ccc} 0 & \nu_{1}+\omega_1 & -\omega_{2}\\ 0 & 0 & 0\\ \nu_2 & 0 & 0 \end{array}\right]\left[\begin{array}{c} c\\ X_{1}^{-}\\ X_{2}^{-} \end{array}\right]\rho_{s}$$ but in the question the form is $$\left[\begin{array}{ccc} c^{\dagger} & X_{1}^{+} & X_{2}^{+}\end{array}\right]\left[\begin{array}{ccc} 0 & \nu_{1} & -\omega_{2}\\ \omega_{1} & 0 & 0\\ \nu_{2} & 0 & 0 \end{array}\right]\left[\begin{array}{c} c\\ X_{1}^{-}\\ X_{2}^{-} \end{array}\right]\rho_{s}$$ and there are 4 noise terms. You could consider some kind of similarity transformation on the $$Q$$ but it seems to me that the transformation would have to be noise dependent.